B*-algebra

Encyclopedia : B : BA : BAL : B*-algebra

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B*-algebras are mathematical structures studied in functional analysis. A B*-algebra A is a Banach algebra over the field of complex numbers, together with a map * : A → A called involution which has the following properties:

1. (x + y)* = x* + y* for all x, y in A.
2. (λ x)* = λ* x* for every λ in C and every x in A; here, λ* stands for the complex conjugation of λ.
3. (xy)* = y* x* for all x, y in A.
4. (x*)* = x for all x in A.
5. ||x*|| = ||x||, i.e., the involution is compatible with the norm.

B* algebras are really a special case of * algebras; a succinct definition is that a B*-algebra is a *-algebra that is also a Banach algebra.

If the following property is also true, the algebra is actually a C*-algebra:

• ||x x*|| = ||x||² for all x in A.

See also: algebra, associative algebra, * algebra.

 

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